We describe totally dissipative parabolic extensions of the one-sided Bernoulli shift. For the fractional linear case we obtain conservative and totally dissipative families of extensions. Here, the property of conservativity seems to be extremely unstable.

The Ito equation is shown to be a geodesic flow of $L^2$ metric on the semidirect product space $\widehat{{\mathrm{𝐷𝑖𝑓𝑓}}^s\left(S^1\right)⨀C^{\infty }\left(S^1\right)}$, where ${\mathrm{𝐷𝑖𝑓𝑓}}^s\left(S^1\right)$ is the group of orientation preserving Sobolev $H^s$ diffeomorphisms of the circle. We also study a geodesic flow of a $H^1$ metric.

Let F be a C ∞ vector field defined near the origin O ∈ ℝn, F(O) = 0, and (Ft) be its local flow. Denote by the set of germs of orbit preserving diffeomorphisms h: ℝn → ℝn at O, and let , (r ≥ 0), be the identity component of with respect to the weak Whitney Wr topology. Then contains a subset consisting of maps of the form Fα(x)(x), where α: ℝn → ℝ runs over the space of all smooth germs at O. It was proved earlier by the author that if F is a linear vector field, then = . In this paper we present...

In this paper, we consider the natural complex Hamiltonian systems with homogeneous potential $V\left(q\right)$, $q\in {ℂ}^n$, of degree $k\in {\mathbb{Z}}^{☆}$. The known results of Morales and Ramis give necessary conditions for the complete integrability of such systems. These conditions are expressed in terms of the eigenvalues of the Hessian matrix $V^{\prime \prime }\left(c\right)$ calculated at a non-zero point $c\in {ℂ}^n$, such that $V^{\prime }\left(c\right)=c$. The main aim of this paper is to show that there are other obstructions for the integrability which appear if the matrix $V^{\prime \prime }\left(c\right)$ is not diagonalizable....

For a class of quadratic polynomial endomorphisms $f:{ℂ}^2\to {ℂ}^2$ close to the standard torus map $(x,y)\to (x^2,y^2)$, we show that the Julia set J(f) is homeomorphic to the torus. We identify J(f) as the closure ℛ of the set of repelling periodic points and as the Shilov boundary of the set K(f) of points with bounded forward orbit. Moreover, it turns out that (J(f),f) is a mixing repeller and supports a measure of maximal entropy for f which is uniquely determined as the harmonic measure for K(f).

We study the jumps of topological entropy for $C^r$ interval or circle maps. We prove in particular that the topological entropy is continuous at any $f\in C^r\left([0,1]\right)$ with $h_{top}\left(f\right)>\left(log⁺\right||f^{\text{'}}{\left|\right|}_{\infty })/r$. To this end we study the continuity of the entropy of the Buzzi-Hofbauer diagrams associated to $C^r$ interval maps.

This paper is concerned with compact Kähler manifolds whose tangent bundle splits as a sum of subbundles. In particular, it is shown that if the tangent bundle is a sum of line bundles, then the manifold is uniformised by a product of curves. The methods are taken from the theory of foliations of (co)dimension 1.

We prove an infinite dimensional KAM theorem which implies the existence of Cantor families of small-amplitude, reducible, elliptic, analytic, invariant tori of Hamiltonian derivative wave equations.

This talk is concerned with the Kolmogorov-Arnold-Moser (KAM) theorem in Gevrey classes for analytic hamiltonians, the effective stability around the corresponding KAM tori, and the semi-classical asymptotics for Schrödinger operators with exponentially small error terms. Given a real analytic Hamiltonian $H$ close to a completely integrable one and a suitable Cantor set $\Theta$ defined by a Diophantine condition, we find a family ${\Lambda }_{\omega },\omega \in \Theta$, of KAM invariant tori of $H$ with frequencies $\omega \in \Theta$ which is Gevrey smooth with...

This paper proposes a stochastic diffusion model for the spread of a susceptible-infective-removed Kermack–McKendric epidemic (M1) in a population which size is a martingale $N_t$ that solves the Engelbert–Schmidt stochastic differential equation (). The model is given by the stochastic differential equation (M2) or equivalently by the ordinary differential equation (M3) whose coefficients depend on the size $N_t$. Theorems on a unique strong and weak existence of the solution to (M2) are proved and computer...